There's also the fact that the new Enterprise is bigger than a Sovereign class, the energy required to beam that much mass would be incredible.

They'd likely have to land her back at the Iowa yards, encase her in something then hook the assembly up to the national grid then beam her somewhere.

Considering with the new drive systems in this universe you can get 4 days travel done in 12 hours, there's not much point wasting 2 or 3 days and a massive amount of energy doing it the other way.

Beaming a Section 31 operative into the Romulan data archives and back out again however...

Yeah, i'm not really buying that. If you can transport a dude from Earth to Qonos with something the size of a duffle bag, you hook that same device up to a matter/anti-matter reactor which gives you huge amounts of power, it should be able to do the same for starship, no problem.